A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Tuesday, August 26, 2008

Today's been a bit better. I've gotten a lot done, both research-wise and teaching-wise, I've met several more of my new students, made a Math Club meeting and a meeting with the high schooler I'm mentoring as she helps me run Super Saturday this semester, and I sat in on Lulabelle's sociology class, Social and Cultural Inquiry.

To me the most beneficial aspect of my sitting in on her classes (and her sitting in on mine...tomorrow it's her turn to come to Precalc) is the chance it gives me to observe many of the same pedagogical techniques I use in my classes in an entirely different disciplinary setting. Many of the same tactics are at play: small group work, writing-to-learn, brainstorming, student discussion. And while the groundwork for Lulabelle's course is entirely different from that of my own, the ends met by the tactics she employs are ultimately very similar: both of us want our students to learn how to think critically, how to assess data and other evidence, how to solve problems and make connections between the distinct and discrete bits of information they pick up here and there.

After 75 minutes had passed, I'd punctuated my notepad with no fewer than nine ?s indicating follow-up questions I had for her when we met after class:

"How do you draw the quiet ones into the discussion? Do you?"

"Have you read The experience economy?"

"Do you like people to raise their hands [before speaking]?"

"Might this work well with small 'break-out' groups to cogitate solutions before reconvening?"

"What's your motivation for reading the daily reading out loud? Is this to achieve a multimodal pedagogical style?"

And so forth. I'm finding our exercise in comparative pedagogical practices very fruitful; it's making me much more aware of my own choices as a teacher. I hope Lulabelle's getting as much out of it as I am.

Monday, August 25, 2008

You ever have one of those days when you just can't seem to get traction?

Today began early, and pleasantly enough. I had a nice walk under cover of slate gray skies, the sun's rise obscured by a feathery blanket of cloud. It was muggy, but comforting.

I'm trying to figure out where my Abstract folks are right now: this morning had me meeting with several of them about the committee problems that were due in class today. A few of them seemed a little overwhelmed, but in class most seemed on the ball. My hunch is that at this point what we're doing doesn't seem relevant, since we're developing skills and techniques, not results that are important per se. Moreover, with the scant background we've developed so far, it's difficult to see how what we're doing fits together with the overall thrust of the course. Once we start doing congruence arithmetic (which we're poised to start discussing on Friday), things will start to come together. In particular, I'll be able to point at congruence classes and say "See? See?!!? Groups!"

From this morning's Abstract section (I'm sorry I was a bit sluggish, y'all...I had a too-short weekend, too!) I went back to the office and took care of a number of bureaucratic tasks...those damned things pop up like whack-a-mole moles at a second-rate funplex. I was busy enough to not even notice the clock inch towards 1:45.

I felt out of sorts in Precalc, too. I think what threw me off at first was my continued frustration over the software. I'm sorry, y'all, that registering on Educo is such a pain. I promise you, I promise you, that once you're all in the lane and plugging away on-line, things will go a bit more smoothly.

My stutter-step start to that class continued, and I felt a half-beat behind for the whole hour. I think part of it's due to my sub-par preparation...I realized about halfway into one of the exercises I'd prepared (describing the interval defined by |x+6|<3) that I hadn't completely prepared a description of the algebraic solution to the problem. That oversight's due in part to my unfamiliarity with teaching this material (again, and again and again: first time doing this, folks!), but part is also due to my abhorrence of "plug 'n' chug" explanations.

Me: "You'll wanna do it this way because if you remember, what the absolute value |a-b| represents is the distance between a and b..."

Not me: "Why do this computation? Because it works, that's why."

Ouch.

I need to find a middle ground.

Happily, I've noted a couple of students who are very attentive, diligent, and responsive to my efforts...they're clearly engaged in class and give great non-verbal feedback as I'm teaching. By watching them I can tell if I seem to be getting through. My thanks to you!

I have to say that I enjoyed my second section of Abstract today, overly large as the section may be. Although there were a few people who I sensed were having difficulties with Euclid's Algorithm, I think most were picking it up, judging from the alacrity with which they offered to share their computations with the class at large.

My grades for the day: Abstract: B+; Precalc: B-; Overall: B.

Bleh.

I'll be ready for Wednesday, folks. You can count on it. Thanks for your patience with me.

Friday, August 22, 2008

Having been doing math "professionally" for quite some time now (long enough that I'm not going to bother counting the years) I tend to forget just how well-built the mathematical machine really is.

This semester's already forcing me to reflect on the inner workings of some simple math concepts that I've sadly learned to take for granted.

Like absolute values: how marvelously they work! You really can depend on |a-b| = a-b to hold as long as a > b. Who'd'a thought? From the textbook I pulled the following absolute value problem with which the students are soon (today or Monday) to grapple: "find a formula for |x+1|-|x-3|, for any x."

Just now I was sitting here at my desk working through the problem so that I know we won't encounter any roadblocks during class...and I was marveling at how well one fares if one forgets everything else one knows about absolute values beyond the its basic defining formula. It's all just adding and subtracting, after all: if x < -1, you get |x+1| = |x-(-1)| = -1-x,and |x-3| = 3-x, so the difference you seek is -1-x-(3-x) = -4. It's marvelous!

This is what I'd hoped teaching this precalculus course would do for me: it's bringing me back into touch with the bits and bobbins at the base of mathematics that I've too long taken for granted. It's making me rethink why one does the things one does, and reflect on what one's really doing when one thinks one is doing something one's not really doing at all.

Thursday, August 21, 2008

I'm lovin' this semester, lemme tell ya. I'm still pretty much on top of the world (a friend called me "Happyman" today).

I spent the day getting ahead in my Precalc notes and further revising my paper with Tip...and meeting with my new students! About my job: I come for the glitz and the glamor, but I stay for the students. These people are great. They're animated, dedicated, smart. It's just one day into the student interviews and I've already scoped out another potential undergraduate research partner and another maybe math major.

167 was a lot more fun today, we actually got to talk about math! If only there were no software glitches. It's proving a pain to get everyone enrolled in the Educo website.

Okay, I'm sorry for cutting this so short again, but I must away to repast.

Wednesday, August 20, 2008

Going straight from 167 to 461 with only a five-minute breather was a pain in the butt. I guess technically my "velocity" was 294/5 = 58.8 course numbering points per minute. Shoot me now.

I thought my first day of Precalc went well...to the extent that I can tell that it really was a Precalc class. I was a little frustrated that explaining the course software and demonstrating its basic functionality to the students takes so damned much time. Fully half of the period today was devoted to pulling the software up, showing students how to log in and pull up the homework assignments, and showing them how to register the software. (It would help matters if there were one way and only one way to register. Grrrrr...)

Tomorrow, folks, tomorrow: I promise we'll start doing math tomorrow. If any of my students from MATH 167 are reading this, another thanks goes to you for slogging through today's bureaucratic rat's nest. I appreciate your attention, and I look forward to working with you.

On the plus side, I believe I managed to nail all but one of the students' names on the first try...at least among those whose names I should have known. 22 out of 23 ain't bad.

From 167 it was a 30-second walk down the hall to the bigger, rowdier section of Abstract. I'm going to need duct tape and thorazine for this crowd, something tells me. They're a lively bunch, but very dedicated to their work, too, and we had some good brainstorms. (The first day's activities asked the students to abstract from the set of integers the essential properties that make this set a group under the operation of addition.) I hope now that everyone's got some sense of a what a group is, so we can promptly push the definition to the back of our minds for a few weeks and work through some preliminary hoo-hah.

I am bushed.

But elated.

It's been a great day. What will tomorrow bring? How could it possibly get better than this? I love my job!

MATH 461 is underway. Today's class went splendidly. The students worked well together, showed only the slightest trepidation about hitting the board first thing out of the gate, and came together to craft a great classroom experience. I'm happy.

I feel ready to face today (it's a beautiful day, by the way), having had the help and support of many wonderful colleagues and friends. In particular, many thanks go to my colleagues Fyodor and Ignatz for helping me to get on the ball with the Educo software for MATH 167, to Lulabelle and Thalia for their perpetual moral support, and to Maggie for everything!

Wednesday, August 13, 2008

Yup, it's that time of year again...Fall's nearly here, classes begin in less than a week, and I'm freaking out that I'm nowhere nearly as well-prepared as I ought to be.

Likely that's not true, but hey.

What's eating at me most this time around is my preparation for Precalc. I've never taught Precalc before (not here, not anywhere...the closest thing I've ever taught was Vandy's one-semester "Survey of Calculus" that took the highlights from a two-semester course of calculus and condensed it into a single semester for the benefit of folks like business majors who didn't want to bother with proofs), for one thing, and for another the course as we teach it involves a substantial on-line homework system with which I have next to no familiarity. It's easy enough to learn, but I'm still puzzling through the details myself and I know it's going to be a few weeks before I'm proficient enough to help guide my students through it.

I'm definitely feeling underprepared. With a week to go, I've got the first couple of days of both of my classes ready: Precalc will start off with a review of the structure of the real numbers, before vaulting into absolute values. Abstract begins with an examination of the Division Algorithm and related results. I'd like to be a week further ahead than this...I've still got next Monday and Tuesday to prepare, though, and with classes starting on Tuesday the hum and thrum of surrounding activity should invigorate me.

Just a couple of days back I was chatting with one of the REU students, and we talked about how we both feed off of the energy of the semester, we get high off of our work and the work that's demanded of us. I'm looking forward to getting that high again.

On a positive note, I've come up with several good ideas for relevant, meaningful Precalc projects that should not only prove challenging and enlightening to the students, but should also give them a leg up when they get to certain cognate problems in their future calculus classes.

For instance, a standard optimization problem in first-year calculus courses is to maximize the volume of a box that can be built from a piece of material of fixed dimensions, according to fixed parameters. The only step that actually involves calculus is the differentiation of the formula the students obtain for volume as a function of one of the linear metrics for the box; deriving that formula in the first place, plotting it and interpreting the graph, and determining maxima and minima visually are all actions that require the skills taught in Precalculus, and students who accomplish such tasks now will be more ready to handle the cognate problems when they recur in the calculus curriculum.

Another plus: while running into campus this morning I finally figured out the general form for the function that governs the work Thalia and I have both kept working on since parting a little over a week ago. I don't know why I didn't see it before, but now with the general equation in mind we might be able to make real progress on actual proofs.

Ah, research.

Here's hoping this semester goes well! Once things get underway, I'm sure I'll slip into the right frame of mind. I always have, for nearly three decades now. I'll be fine. Besides, I've already got pledges of moral support from two dear friends should Precalc prove to be the back-breaking straw.

Much as I know I need to prepare a bit more, I'm having difficulty focusing. I'm feeling very sluggish today. And why should I not be? The weather took a much-needed turn for the rainy overnight, so the sky is a shroud of white. Campus (where I sit as I type this) is dead: my department's a tomb, the building's air is still and warm (the cooler's on the fritz...again), all is quiet.

All right, I'm going to head home...no sense in cocooning myself in this scholarly sarcophagus any longer today. I'm sorry this post has been so scattered and erratic.

Sunday, August 10, 2008

I've had a week now to recover from the departure of my summer's colleagues, the wonderful students that made up my REU team for the past eight weeks. The last of them took flight almost exactly a week ago this moment, winging her way back to New York last Sunday afternoon.

I miss them. They're great students, smart colleagues, and close friends. I look forward to seeing them at conferences, working with them on papers, growing up and learning by their sides.

For now they're gone, but the work continues: with two of them (those with whom I worked most closely with) I've already exchanged write-ups, graphs, ideas...yes, work continues.

But the cycle's begun anew: I arrived at my office at 7:15 this morning to find a note left on my door by two of my favorite students from last year, informing me that they're back (for RA duties, no doubt) and were surprised that they'd missed me. They were operating under the (not-all-that-erroneous) assumption that I'm never more than a hundred yards or so away from my office.

I spent this morning helping first-year students move in. For three hours I shuttled luggage carts back and forth between parents' cars and dorm rooms, schlepping all manner of stuff into Mills Hall. Fully laden one way, emptied out on return, several trips for several students. One's a budding music major, another's contemplating environmental science. A few are undecided: biology? History? "I think I want to teach."

"Math?"

"No! I'm not really good at math."

"That's what everyone says until they find out it's not true," I assured her.

It was refreshing, revitalizing. Its sense of newness and rebirth stood in contrast to the moving out my REU students did the weekend before.

My move into Johnson-MacFarlane Hall at the University of Denver 15 years (!) ago was nothing like this: my dad parked his truck in the middle of the phalanx of vehicles that lines that lined High Street for a block in either direction, and as soon as the dorm's outside doors were unlocked students and parents descended on the dorm like sharks in a frenzy. It was chaotic. There were no faculty or staff helping people to move in, there wasn't even anyone to direct activities or answer simple questions (where's the bathroom? Is the dining hall open? Where's the nearest ATM?)...that day I saw my RA cry for the first of what would be many times that year. (She was a gentle soul and didn't deserve some of the folks who lived on our floor.)

This coming week will see me getting the first few days of my two classes ready. I've read through the first few chapters of the text I'll be using for Abstract Algebra, and I've brainstormed a few opening exercises for that course, but I've yet to commit anything to paper. Tomorrow I'm meeting with one of my colleagues who's taught Precalc a number of times, and I'm hoping he can then give me a rundown on the topics I should be sure to address during the next fifteen weeks.

I've also got to get some more work done on the presentation on the writing program I'll be making at the Carolinas Writing Program Administrators conference in a month or so. Lulabelle and I met late last week to discuss that issue, but we spent most of our meeting chatting and going over the draft of my math poetry paper and syllabi.

Much work to be done, I just don't feel like doing it right now.

Time's running out, though...not long before classes start, and then where will I be?

Argh.

Okay, I'm off...I feel like I've got a lot to say, but so few words with which to say it. I've got to gather them together and compose my self before I can make any more sense.

Saturday, August 02, 2008

As you may surmise from my four-week blogospheric absence (an absence that has elicited chagrin from at least a small handful of dedicated readers...most of them connected to the primary reason for my absence in the first place), things have been a mite busy in my neck of the woods.

The program's nearly complete: four have now left, four remain.

Yesterday was the last "official" day of the 2008 REU, on which the students gave their final presentations and partook of a summative ceremonial repast at the West Asheville Asiana Grand Buffet on Smoky Park Highway.

Huzzah.

The presentations were wonderful, truly of a quality greater than that of most faculty conference talks. Although a few of the kids were obviously nervous, they did marvelously.

I'm stunned by the caliber of research they've produced, as well. The new knowledge they've brought into the world should make fodder for at least four or five papers, and all of the projects they've begun are worth following up. Some have barely started! I'll probably spend a good deal of time in the next few weeks beginning my share of the papers that should come from the program.

The better part of my time, however, will be dedicated to getting ready to teach this coming term's courses: Precalc and Abstract Algebra. I've not taught Abstract here before (I did a semester in Illinois), so that'll be something new...and I've never taught Precalc...nowhere, no how. I'm looking forward to that experience. I've already met a few of my students (they're friends of past students), and they seem a good lot. It's going to be a departure for me content-wise, but a worthy challenge pedagogically.

Elsewhere in my professional career, I've now written about four or five pages of my paper on math poetry. I've yet to really pick apart many of my students' responses to the experience of writing math-themed poetry, aside from identifying a "sense-making" trend in their poems (about which I believe I may have blogged before). My reflections on the students' reflections should form a core part of the paper, and I hope to get to that soon.

I've also got two papers to referee and one to review...reading material for the beach when Maggie and I head down to Savannah for a few days at the end of the week after next.

Ah, well.

I'm going to go now and have a bite of breakfast. I plan to return more soon than four weeks hence, so stay tuned.